So on the surface this is just some collection of really ****ty, unexceptional pepes. But it has some relation to bitcoin? If there's something beyond just a ****ty database going on here I can't understand it at all. Could someone who knows anything about toasters please explain?

They're using the blockchain to try and bypass the fact that Pepes are infinitely reproducible and enforce scarcity. There's (also?) a meme-based cryptocurrency version but there are plenty of those already.

It's all wholesome internet meme fun until someone comes along and uses it for money laundering.

Blockchain is a distributed ledger recording all transactions involving the cryptocurrency. It allows bitcoin and other currencies to function without a central authority recording transactions because the ledger is updated any time currency is transferred from one wallet to another.

Instead of currency, those guys are using it to determine ownership of the rare Pepes they've assembled. They'll run into trouble though, because even if their Pepe's are technically unique it doesn't matter how rare a Pepe is if it isn't high-quality.